Chapter 5 The Idea of Probability

1. Chapter 5The Idea of Probability AP Statistics

2. The Point of this Chapter • When we do an experiment or observational study, we would like to know if our results are statistically significant. • That is, we want to know if the results we obtained were likely to occur simply by chance. • To determine if our results are statistically significant, we need calculate the probability, which is what we will study in the next chapter.

3. Definition of Probability • The study of probability is the systematic study of uncertainty. • Probability is the long run relative frequency of an occurrence. • A chance experiment is any activity or situation in which there is uncertainty about which of two or more possible outcomes will result. • ex: flipping a coin, rolling a die, choosing a card, asking a survey question, giving a treatment

4. Understanding Probability • Anevent is any collection of outcomes from the sample space of a chance experiment. • Note: We usually denote events with capital letters: A, B, C, ...or a capital letter with subscripts: • ex: A = choosing a senior male = {senior male from 2nd period, senior male from 3rd period or senior male from 4th period} • A simple event is an event consisting of exactly one outcome. • ex: B = choosing a junior female from 3rd period = {junior female from 3rd period} • ex: find an event C such that C has three outcomes. List the outcomes.

5. Understanding Probability • The probability that an event will occur is expressed as a number between 0 and 1. • The following are examples of the notation we use for probability: • P(A) • represents the probability that the event A will occur • P(A) = 1 • represents the probability that event A will occurs is 1 or that the probability that event A occurs is CERTAIN • P(A) = 0 • represents the probability that event A will occurs is 0 or that the probability that event A occurs is IMPOSSIBLE

6. Understanding Probability • The collection of all possible outcomes of a chance experiment is the sample space for the experiment. • ex: flipping a coin: heads, tails • ex: rolling a die: 1, 2, ... 6 • ex: choosing a card: ace of hearts, 2 of hearts, .....king of spades (52 total)

7. Understanding Probability • In some chance experiments, more than one piece of data is collected. • ex: a randomly selected stats student will be asked for his or her gender, class, and period. • Thus, one possible outcome would be a Senior Male in 3rd period. • We can use a tree diagram, with each set of branches corresponding to one variable and each “end” representing one outcome. • How many possible outcomes are there? • To identify all possible outcomes, we may wish to use the Fundamental Counting Principle: • If there are n ways that Event A can occur and m ways that Event B can occur, then there are m*n ways that both can occur

8. More Definitions • For any random phenomenon, each attempt, or trial, generates an outcome. Combinations of outcomes are called events. • Once again, a phenomenonconsists of trials. Each trial has an outcome. Outcomes combine to make events. • Things are simplified as long as each trial is independent– generally speaking, this means that one trial doesn’t affect the outcome of another.

9. Always? • Will the long run average always give us a regular distribution? • Yes. In fact, this is one of the most important concepts to probability, a principle called the Law of Large Numbers (LLN). • The Law of Large Numbers says that as the number of trials increases, the relative frequency of an event will approach the true probability of the event • The LLNguaranteesthat the long-run relative frequency of repeated independent events settle down to the true probabilityas the number of trials increases.

10. The LLN and the Gambler’s Fallacy • People often misunderstand or misuse the Law of Large Numbers because the idea of the long run is so abstract. • Many people often think that after seeing a series of similar events (such as six heads in a row when flipping a coin) suggests that an alternate event is “due” (after six heads, there is a greater chance of getting tails since it is “due”) – however this is an error called the Gambler’s Fallacy.

11. The LLN and the Gambler’s Fallacy • The problem is that the LLN doesn’t apply in short-run behavior. The probability only evens out in the long-run! In fact, it must be infinitely long before we are sure to get the true probabilities. • Just because you flip a coin and get heads six times in a row, doesn’t mean that a tails is due. The coin doesn’t hold a memory and make things come out right! • In fact, in a very long series, you would see many streaks of heads, but afterwards, there does not need to be a streak of tails.

12. A Strategy for Winning the Lottery • A common proposal for beating the lottery is to note which numbers have come up lately, eliminate those from consideration, and bet on numbers that have not come up for a long time. Proponents of this method argue that in the long run, every number should be selected equally often, so those that haven’t come up are due. • Is this a good strategy…Why or why not?

13. A Strategy for Winning Keno • A group of graduate Statistics students decided to take a trip to Reno. They very discretely wrote down the outcomes of the game for a couple of days (several hundred runs of the game), then they drove home and tested whether or not the numbers were equally likely. It turned out that some numbers showed up more often than others. Rather than betting on the numbers that were “due,” they conjectured that according to the LLN, certain numbers were more likely than others and betted on those numbers. They took $5000 and put their theory to the test.

14. A Strategy for Winning Keno • Did the Statistics students have a good strategy? • Why or why not? • So what happened? • The students ended up pocketing over $50,000. They were escorted off the premises, and asked to never show their faces in that casino again. • Why was the first example an instance of the gambler’s fallacy and the second example a proper use of the LLN?

15. Probability • Finding probabilities are easy as long as each event is equally likely. Be careful when they are not. Consider the following interview: • Interviewer: What do you think your chances are of winning the lottery? • Lottery ticket purchaser: Oh, about 50-50. • Interviewer: What?! How did you get that? • Lottery ticket purchaser: Well, I figure that either I win or I don’t! • The moral of the story is that outcomes are not always equally likely!